Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  elpcliN Structured version   Unicode version

Theorem elpcliN 35090
Description: Implication of membership in the projective subspace closure function. (Contributed by NM, 13-Sep-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
elpcli.s  |-  S  =  ( PSubSp `  K )
elpcli.c  |-  U  =  ( PCl `  K
)
Assertion
Ref Expression
elpcliN  |-  ( ( ( K  e.  V  /\  X  C_  Y  /\  Y  e.  S )  /\  Q  e.  ( U `  X )
)  ->  Q  e.  Y )

Proof of Theorem elpcliN
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 simp1 996 . . . . . 6  |-  ( ( K  e.  V  /\  X  C_  Y  /\  Y  e.  S )  ->  K  e.  V )
2 simp2 997 . . . . . . 7  |-  ( ( K  e.  V  /\  X  C_  Y  /\  Y  e.  S )  ->  X  C_  Y )
3 eqid 2467 . . . . . . . . 9  |-  ( Atoms `  K )  =  (
Atoms `  K )
4 elpcli.s . . . . . . . . 9  |-  S  =  ( PSubSp `  K )
53, 4psubssat 34951 . . . . . . . 8  |-  ( ( K  e.  V  /\  Y  e.  S )  ->  Y  C_  ( Atoms `  K ) )
653adant2 1015 . . . . . . 7  |-  ( ( K  e.  V  /\  X  C_  Y  /\  Y  e.  S )  ->  Y  C_  ( Atoms `  K )
)
72, 6sstrd 3519 . . . . . 6  |-  ( ( K  e.  V  /\  X  C_  Y  /\  Y  e.  S )  ->  X  C_  ( Atoms `  K )
)
8 elpcli.c . . . . . . 7  |-  U  =  ( PCl `  K
)
93, 4, 8pclvalN 35087 . . . . . 6  |-  ( ( K  e.  V  /\  X  C_  ( Atoms `  K
) )  ->  ( U `  X )  =  |^| { z  e.  S  |  X  C_  z } )
101, 7, 9syl2anc 661 . . . . 5  |-  ( ( K  e.  V  /\  X  C_  Y  /\  Y  e.  S )  ->  ( U `  X )  =  |^| { z  e.  S  |  X  C_  z } )
1110eleq2d 2537 . . . 4  |-  ( ( K  e.  V  /\  X  C_  Y  /\  Y  e.  S )  ->  ( Q  e.  ( U `  X )  <->  Q  e.  |^|
{ z  e.  S  |  X  C_  z } ) )
12 elintrabg 4301 . . . . 5  |-  ( Q  e.  |^| { z  e.  S  |  X  C_  z }  ->  ( Q  e.  |^| { z  e.  S  |  X  C_  z }  <->  A. z  e.  S  ( X  C_  z  ->  Q  e.  z )
) )
1312ibi 241 . . . 4  |-  ( Q  e.  |^| { z  e.  S  |  X  C_  z }  ->  A. z  e.  S  ( X  C_  z  ->  Q  e.  z ) )
1411, 13syl6bi 228 . . 3  |-  ( ( K  e.  V  /\  X  C_  Y  /\  Y  e.  S )  ->  ( Q  e.  ( U `  X )  ->  A. z  e.  S  ( X  C_  z  ->  Q  e.  z ) ) )
15 sseq2 3531 . . . . . . . 8  |-  ( z  =  Y  ->  ( X  C_  z  <->  X  C_  Y
) )
16 eleq2 2540 . . . . . . . 8  |-  ( z  =  Y  ->  ( Q  e.  z  <->  Q  e.  Y ) )
1715, 16imbi12d 320 . . . . . . 7  |-  ( z  =  Y  ->  (
( X  C_  z  ->  Q  e.  z )  <-> 
( X  C_  Y  ->  Q  e.  Y ) ) )
1817rspccv 3216 . . . . . 6  |-  ( A. z  e.  S  ( X  C_  z  ->  Q  e.  z )  ->  ( Y  e.  S  ->  ( X  C_  Y  ->  Q  e.  Y ) ) )
1918com13 80 . . . . 5  |-  ( X 
C_  Y  ->  ( Y  e.  S  ->  ( A. z  e.  S  ( X  C_  z  ->  Q  e.  z )  ->  Q  e.  Y ) ) )
2019imp 429 . . . 4  |-  ( ( X  C_  Y  /\  Y  e.  S )  ->  ( A. z  e.  S  ( X  C_  z  ->  Q  e.  z )  ->  Q  e.  Y ) )
21203adant1 1014 . . 3  |-  ( ( K  e.  V  /\  X  C_  Y  /\  Y  e.  S )  ->  ( A. z  e.  S  ( X  C_  z  ->  Q  e.  z )  ->  Q  e.  Y ) )
2214, 21syld 44 . 2  |-  ( ( K  e.  V  /\  X  C_  Y  /\  Y  e.  S )  ->  ( Q  e.  ( U `  X )  ->  Q  e.  Y ) )
2322imp 429 1  |-  ( ( ( K  e.  V  /\  X  C_  Y  /\  Y  e.  S )  /\  Q  e.  ( U `  X )
)  ->  Q  e.  Y )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   A.wral 2817   {crab 2821    C_ wss 3481   |^|cint 4288   ` cfv 5594   Atomscatm 34461   PSubSpcpsubsp 34693   PClcpclN 35084
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6298  df-psubsp 34700  df-pclN 35085
This theorem is referenced by:  pclfinclN  35147
  Copyright terms: Public domain W3C validator